Question

2. Consider penalty shoot-outs in soccer that happen when the score is tied after the regulation playing time'has expired (Let's call this state "tie"] Suppose that there two teams (Team A and Team B), and rounds of one kick per team, until the tie is broken (that is, starting from a tied score, there is a round of one penalty kick per team. If Team A scores, but team B doesn't, then Team A wins the game. But if both teams score (or neither of them scores), then we're again at a tied score, and there's an additional round of one penalty kick per team, and so on). Team A takes the first kick in the first shoot-out. Suppose that both Team A and Team B could make a penalty kick with the probability of 0.8. The winning team gets $10, and the losing team pays $10. a. Is this game recursive? Draw the game tree. If there is a state that recurs, do not go beyond that state. b. What is Team A's probability of winning the game in the state of tie? c. d. What is the probability of winning the game for Team A given it doesn't score on its first penalty kick? What is the value of the game? e.

Answer 1

a) Yes this game is recursive. b) The game tree is given like the following:

0 eoM 0 = Loss

c) The probaility of team A winning after a tie is: - P(Winning after 1st tie) + P(Winning after 2nd tie) P(Winning after 3rd tie) + [(0.8)2 (0.8) (0.8) ...] (0.8) x (0.2(0.2) (0.2)(0.2) (08)2-(0.8) × (0.2) + 0.2ア 1- (0.8)2 (0.8) x (0.2) (0.2)2 (0.8) (0.2) 0.29 d) The probability of team A winning and not scoring a penalty in the 1st kick is P(Making a tie while not scoring at first and winning after that) + P(Making 2 ties while not scoring at first and winning after that) P(Making 3 ties while not scoring at first and winning after that) +... (0.2)2 × 1)(The probaility of tean A winning after a tie) 0.04 × 0.29 Hence, by Bayes' Theorem P(The probability of team A winning|Not scoring a penalty in the 1st kick) P(The probability of team A winning and not scoring a penalty in the 1st kick) P(Team A not scoring a penalty in the 1st kick) 0.04 × 0.29 0.2 0.29 0.058 0.2 e) Observe that Probability of Team 2 winning is also the same as Team A because of the same probability of getting a penalty score. Hence the probaility of team A (or B) winning is - P(Team A winning in 1st round) + P(Team A winning after a tie) 0.8 x 0.20.29 0.16 0.29-0.45 Therefore, the value of the game is $10 x 0.45+ $(-10) x 0.45$0 S0